I was pretty happy with how this week went. I finished the unit Composite Functions on Wednesday and got halfway through Trigonometry before the week ended. As I hoped, I was able to make a lot more sense of trig this week than I was able to when I first learned it. It did take me a bit of time to connect a few of dots but, overall, the concepts that I found hard to understand a few months ago seemed fairly straightforward this week. Considering how long I worked on trig and how tough it seemed at the time, it was a great feeling being able to make sense of it all and finally having everything ‘click’. Thinking about how simple some of the concepts seem now, it’s funny how difficult and frustrating trig was when I first learned it. It makes complete sense, however, and follows the same pattern of everything I’ve learned in KA – new concepts are initially hard to grasp and then, in retrospect, they always simple and obvious. This was a good reminder for me as I move forward into calc.

I had a “duuuhhhh” moment halfway through the week when I finally understood what the inverse of number is. Here’s a picture of my note that I wrote when I finally understood it:

For some reason I always had in my mind that ‘inverse’ meant ‘opposite’ which I for some reason thought meant the inverse of a positive number would be the same number but negative, ex. the inverse of 3 would be -3. When I worked on invertible functions this week where I raised numbers to negative exponents, ex. 4^-1, it finally made sense that the negative exponent moves the base, 4, to the denominator, i.e. 1/4^1, which turns the value into a fraction/decimal between 0 and 1, i.e. 0.25. The key concept that clicked for me is that the inverse of a number >1 is always a decimal between 0 and 1. It’s hard to explain, but I realized that, in the same way you can count from 1 to infinity, there are also an infinite amount of numbers between 0 and  1 which, although I can’t put into words why it’s important, I think is a key part in understanding what an inverse is. 

Getting through the unit Composite Functions was not as difficult as I thought it would be. The only thing I had left to get through were two sections on inverse and invertible functions. These two terms turn out to be two sides of the same coin, so to speak. Here’s a page from my notes that explains the gist of what an inverse function is and, below it, an example picture I found online that is a visual representation of an inverse function:

To understand what an invertible function is you have to remember that when you input a number into the domain of a function, depending on the function, it can map to ONE-OR-MORE values in the range but, going the other direction, you can only map each value in the range to ONE value in the domain. An invertible function, however, is a function that when you input a number into the domain it ONLY maps to ONE number in the range. If a function has parts of it where the domain can map to two-or-more values in the range, it’s not invertible. You can determine if a function is invertible by putting one function into the other, simplifying the expression, and if the result leaves you with the variable used for the x-axis then the function is invertible. Here’s a photo from my notes that works through this process:

In the photo you see that when you put g(x) inside of f(x) and simplify you’re left with x meaning these functions are the inverses of each other and, therefore, that you would say each function is “invertible”. The hardest part I found when solving these questions, which applied to the unit test as well, was following the proper BEDMAS order of operations which I realized I’m a bit rusty at. It was good practice working through these questions however and, although the unit test took me a bit longer than usual, I managed to get 100% on my first attempt.

The first thing I reviewed when I began Trigonometry on Friday was sinusoidal equations and how they relate to the unit circle. Here’s a photo from my notes that shows a basic example of f(x) = sin(x) and a screen shot of the function below it:

I had another “duuuhhh” moment when I realized that the x-axis always measures radians. This means, for example, if you input x = 12 then:

• 12/ π = ~3.8197
  • ~3.8197 * 180° = ~687.549°
    • A.k.a. where x = 12, you’ve gone counterclockwise around the unit circle ~687.549°, i.e. just under two revolutions.

Understanding this, I can now FINALLY visualize how and why sinusoidal functions work. I fully understand why the functions are shaped like a wave, why the period gets shortened, extended and shifted when you multiply, divide and/or add or subtract values to x inside sin(x) or cos(x), how and why the amplitude is stretched or shrunk by putting a number in front of sin(x) or cos(x), and why the function moves up or down the Y-axis when you add or subtract a number to the equation. Being able to understand how/why each of these transformations takes place is one of the most gratifying things I’ve learned on KA so far. 

The last big breakthrough I had this week was getting a way better understanding of what trig ‘identities’ are. Here are two pages from my notes that talks about it:

These pages are tough to understand but the key points are:

• In quadrant (1) of the unit circle:
  • Sin(θ) = y-coordinate = b
  • Cos(θ) = x-coordinate = a
    
• To get the same angle relative to the X-axis in each of the other three quadrants (i.e. a and b have the same absolute value) you add or subtract θ and/or π:
  • Quadrant (2) = (π – θ)
  • Quadrant (3) = (π + θ)
  • Quadrant (4) = (–θ)
    
• Using these angles for quadrants 2, 3, and 4, trig identities simply state that you keep the same absolute values of a and b but change the (+/–) as follows:
  • Quad (1) to quad (2) = b/a changes to b/–a = –(b/a) 
  •  Quad (1) to quad (3) = b/a changes to –b/–a = b/a
  • Quad (1) to quad (4) = b/a changes to –b/a = –(b/a)

In my mind, trig identities simply mean that by using the specific angles (π – θ), (π + θ), and (–θ) you know that the absolute values for x and y will stay they same but the (+/–) will change depending on what quadrant you move to.

This coming week, Week 75 (!), I’d like to watch all videos in the two remaining sections of the unit Trigonometry (600/600 M.P) and get started on the following unit Vectors (0/1400 M.P.) by Thursday. Considering the size of Vectors, I’d be happy if I was able to get halfway through the unit by the end of the week but, having no idea what vectors are, I could see that being a stretch. After I get through Vectors, I’ll have ~3000 M.P. left to go in the course and I’m really hoping I can get through it by the end of February. To be fair, I technically haven’t even started calculus since this course is just Precalculus so it will be nice when I finally get there. One small, agonizingly slow step at a time.